Sorry if this has already been covered, but have you looked into Working Holiday Visas in Australia or New Zealand?
If you won't have bags of cash to fund your travel, this could be a way to get a foothold in another country. Most WHV work is casual and basic, but as i understand it there's nothing to stop you working at your career, and ending up getting sponsored by your employer for a permanent stay if you decide you're happy.
Or the usual casual WHV work does at least give you that "moving from place to place" that you wanted, for a year in each country. (Can be extended to 2 years if you do enough farm work. )
[ Edit: Edited on 13-Jan-2016, at 08:27 by Andyf ]
Working in Australia and New Zealand is an option. I met quite a few foreign workers there last year, particularly in New Zealand. Since you're an American, there is a different, though similar, program than the Working Holiday Visa in Australia: http://www.australianvisaapplications.com/Australia-working-holiday-visas.html. Note the requirements.
Note that the site berner256 links to above is an unofficial third party website; I'm not spotting any major errors on it at first glance, but for visa details, I recommend always going to the official site, in this case that of the Australian Department of Immigration: Work and Holiday Visa (that's the version for Americans). Information about the New Zealand version can be found here.
Thanks, Sander. I wasn't aware that it was a third-party Web site, as I did a quick read (I'm traveling). It's always best to go to the official Web site. While I was in Australia last year, the government announced plans to make changes in taxation related to income received under the WHV program. I don't know if they were implemented.